Portfolio optimisation beyond semimartingales: shadow prices and fractional Brownian motion

TL;DR

This paper demonstrates that for certain non-semimartingale price models, including fractional Brownian motion, one can find a shadow price process within the bid-ask spread that allows for optimal utility maximization under transaction costs.

Contribution

It establishes the existence of shadow prices for non-semimartingale models, extending utility maximization theory beyond classical semimartingale frameworks.

Findings

Shadow prices exist for non-semimartingale models with transaction costs.

In the fractional Brownian motion case, the shadow price is an Ito process.

Fractional Brownian motion trajectories can touch Ito processes without reflection.

Abstract

While absence of arbitrage in frictionless financial markets requires price processes to be semimartingales, non-semimartingales can be used to model prices in an arbitrage-free way, if proportional transaction costs are taken into account. In this paper, we show, for a class of price processes which are not necessarily semimartingales, the existence of an optimal trading strategy for utility maximisation under transaction costs by establishing the existence of a so-called shadow price. This is a semimartingale price process, taking values in the bid ask spread, such that frictionless trading for that price process leads to the same optimal strategy and utility as the original problem under transaction costs. Our results combine arguments from convex duality with the stickiness condition introduced by P. Guasoni. They apply in particular to exponential utility and geometric fractional Brownian motion. In this case, the shadow price is an Ito process. As a consequence we obtain a rather surprising result on the pathwise behaviour of fractional Brownian motion: the trajectories may touch an Ito process in a one-sided manner without reflection.